Framework Flexibility and the Negative Thermal Expansion Mechanism of Copper(I)Oxide, Cu2O

Leila H. N. Rimmer,1 Martin T. Dove,1, 2, 3, ∗ Bjorn Winkler,4

Dan J. Wilson,4 Keith Refson,5 and Andrew L. Goodwin6

1Department of Earth Sciences, University of Cambridge, Downing Street, Cambridge CB2 3EQ, U.K.2Centre for Condensed Matter and Materials Physics, School of Physics and Astronomy,

Queen Mary University of London, Mile End Road, London E1 4NS, U.K.3Materials Research Institute, Queen Mary University of London, Mile End Road, London E1 4NS, U.K.

4Geowissenschaften, Goethe-Universitt, Altenhoeferallee 1, D-60438 Frankfurt a.M., Germany.5Science and Technology Facilities Council, Rutherford Appleton Laboratory,

Harwell Science and Innovation Campus, Didcot OX11 0QX, U.K.6Inorganic Chemistry Laboratory, University of Oxford, South Parks Road, Oxford OX1 3QR, U.K.

(Dated: October 29, 2018)

The negative thermal expansion (NTE) mechanism in Cu2O has been characterised via mapping ofdifferent Cu2O structural flexibility models onto phonons obtained using ab-initio lattice dynamics.Low frequency acoustic modes that are responsible for the NTE in this material correspond tovibrations of rigid O–Cu–O rods. There is also some small contribution from higher frequency opticmodes that correspond to rotations of rigid and near-rigid OCu4 tetrahedra as well as of near-rigidO–Cu–O rods. The primary NTE mode also drives a ferroelastic phase transition at high pressure;our calculations predict this to be to an orthorhombic structure with space group Pnnn.

PACS numbers: 62.20.de, 63.20.-e, 65.40.De

I. INTRODUCTION

Negative thermal expansion (NTE) is a property ofsignificant interest to those working in the field of ma-terials design. Not only does it offer the prospect ofstructures that remain undistorted over significant tem-perature ranges1–5 but, in addition, other unusual phe-nomena (such as softening under pressure or enhancedgas adsorption and filtration) are often linked to the mi-croscopic origins of NTE.6,7 Understanding these micro-scopic driving mechanisms in NTE materials is thereforeof fundamental importance.

FIG. 1. The Cu2O crystal framework. The structure is shownin ball and stick representation: red atoms are O; blue atomsare Cu. One of the two sublattices is shaded in pale grey forclarity.

Copper oxide, Cu2O, has NTE at low temperatures.Its coefficient of thermal expansion has been variouslymeasured as having a minimum value between −8.0 and−2.95 MK−1 at temperatures between 80 and 100 K,before increasing to become weak positive thermal ex-pansion above 200 K.8–11

Cu2O is also one of the simpler NTE-exhibitingframeworks,12 its structure consisting of two interpen-etrating diamondoid cristobalite lattices—see Figure 1.Despite its structural simplicity compared to other NTEframeworks, the atomic-scale origin of the NTE in Cu2Ois not fully understood.

Inelastic neutron scattering,13 Raman scattering andluminescence14 as well as ab-initio calculations13,15 haveidentified specific phonon modes responsible for drivingNTE in Cu2O. All such modes exist below 5 THz, withthe largest contribution coming from transverse acousticmodes in the Γ–X–M region of reciprocal space13,15 (inthis paper we use the notation of Bradley and Cracknell16

to label the high-symmetry points of the reciprocal cell).Optic modes that contribute weakly to both NTE andpositive thermal expansion (PTE) also exist in this en-ergy range.13–15

A common mechanism for NTE suggested for frame-work materials is the ‘tension effect’,17 in which trans-verse vibrations of bridging atoms in a three-atom con-nection bring the two end atoms towards each other dueto the relatively high energy cost of stretching inter-atomic bonds.1,4,18 However, there is a similar mecha-nism in which a group of atoms rotate as a whole andbring inwards the planes of atoms connected to theirends. Both mechanisms are shown in Figure 2. A similarmechanism is found to be important in the related NTEmaterial Zn(CN)2.19 Both types of motions will be cor-

arX

iv:1

402.

1026

v1 [

cond

-mat

.mtr

l-sc

i] 5

Feb

201

4

2

d d’d’

FIG. 2. Illustration of two transverse vibrations that can con-tribute to NTE. Both lead to negligible stretching of nearest-neighbour bonds, but result in reduction of the inter-atomicor inter-planar distances (in this case reducing distance d tod′). If enough of these vibrations exist with large enoughamplitude in a material, they can drive macroscopic NTE.

related and described in terms of phonon normal modes.For zero wave vector, the traditional tension mechanismwill be described by optic phonons, but the rotationalmotion will more likely be part of a transverse acousticphonon that generates a shear deformation of the struc-ture.

All materials undergo some drive towards PTE dueto longitudinal bond expansion arising from the an-harmonicity of interatomic potentials.20 However, if atension-effect mode has low enough energy and thus largeenough amplitude (given that a phonon’s amplitude isinversely proportional to the square of its frequency20)and if it comprises sufficiently large a fraction of the fullphonon spectrum, then the PTE is outweighed and thenet outcome is macroscopic NTE.4

Parametrised potential calculations21 have identifiedan optic phonon that dominates the low energy densityof states. Analysis of its eigenvectors at the Γ point21,22

show this mode to consist of rotations of undeformedOCu4 tetrahedra. This is a type of tension effect knownas a rigid unit mode23 (RUM), an established NTE mech-anism in some silicate minerals.24,25 The same mecha-nism has also been proposed as a result of evidence fromneutron powder diffraction9 where anisotropy of Cu ther-mal displacement parameters points to significant Cu mo-tion transverse to the 〈111〉 directions.

Meanwhile EXAFS work,11,12,26–29 as well as X-ray11

and neutron30 PDF studies, have led to proposals of analternative tension effect mechanism involving significantOCu4 tetrahedral deformations, either in addition to theRUM11 or replacing it.29,31 This is due to observations ofboth expansion and contraction in average Cu· · ·Cu dis-tance as a function of increasing temperature, interpretedas deformations of the OCu4 tetrahedral units.

However, no proposed mechanism has been tested

TABLE I. Cu2O equilibrium cell parameters, as obtainedthrough calculation and experiment. Our value of a = 4.358 Ais a ∼2% overestimation of the experimentally-derived values,a typical result for GGA calculations. The symbol † indicatesthat the measurements were performed at 300 K. The symbol‡ indicates that the measurements were performed at 15 K.

Source a0 / A

This work (PBE+pseudopotentials) 4.358

PBE+pseudopotentials32 4.359

LDA+pesudopotentials32 4.221

PBE+mixed-basis pseudopotentials13 4.30

All-electron calculation13 4.32

X-ray powder diffraction29† 4.27014(7)

Neutron powder diffraction9‡ 4.2763(2)

against the known NTE phonons in Cu2O. It is notenough to determine that a type of large amplitude struc-tural deformation (and therefore possible tension effectmechanism) takes place in the system; in order to under-stand the driving force behind its NTE, we must find thespecific type of tension effect (if any) that corresponds toits NTE-driving modes.

Our approach to this problem is to map phonons fromdifferent models of Cu2O structural flexibility onto thosefrom high quality ab-initio calculations. By observing thedegree to which each flexibility model is able to describethe real system’s NTE modes we can thus determine thetype of framework flexibility, and the particular mani-festation of the tension effect, that drives the NTE inCu2O.

II. METHODS

A. Ab-initio lattice dynamics

Density functional theory33,34 simulations wereperformed using the plane-wave pseudopotentialmethod35 as implemented in CASTEP.36 The GGA-PBE functional37,38 was used for all calculations.

Troullier-Martins norm-conserving pseudopotentialsdescribing Cu and O were obtained from the ABINITFHI database.39,40 The density mixing procedure, with20 densities stored in its history, was implementedfor SCF minimisation. A plane-wave cutoff energy of1100 eV and corresponding FFT grid were used to definethe basis for the electronic orbitals. To minimise viola-tion of the acoustic sum rule arising from high-frequencycomponents of the GGA XC functional, a grid 2.5 timesdenser in each direction was used to represent the elec-tron density and potentials. Electronic Brillouin-zone in-tegrals were evaluated using a 12 × 12 × 12 Monkhorst-Pack grid. In addition, Cu2O was explicitly defined as azero-temperature insulator by keeping band occupanciesfixed.

3

TABLE II. Cu2O phonon frequencies at the Γ point for the equilibrium Cu2O cell volume, as obtained through calculationand experiment. Frequencies are in units of cm−1. Calculated frequencies are within a few percent of values obtained viaexperiment, a typical result for DFT. The symbol † indicates data obtained via Reference 22.

Study T2u Eu T1u (TO) T1u (LO) Bu F2g F1u (TO) F1u (LO)

This work (DFPT calculation) 64 77 139 140 337 489 607 629

DFPT calculation13† 72 86 147 148 337 496 609 629

All-electron finite displacement calculation22 67 119 142 146 350 515 635 654

Raman+luminescence14 86 110 152 152 350 515 633 662

A geometry optimisation was performed at 0 GPa suchthat residual stresses on the cell were within 0.02 GPa.Since the atomic coordinates of Cu2O are fixed by itsPn3m symmetry, only the cell parameter required op-timisation. Table I shows the converged result of a =4.358 A. This is a 2% overestimation of the experimentalvalue, as is typical for GGA-PBE.

Phonons were calculated for a total of six unit celllengths: at equilibrium as well as values of 4.25, 4.30,4.35, 4.40 and 4.45 A. Density functional pertubationtheory41 (DFPT) was used with an 8× 8× 8 Monkhorst-Pack phonon wave vector grid that was offset to placeone of the grid points at Γ. Convergence tolerance forforce constants during the DFPT calculations was set at10−6 eV A−2. Long-range electric field effects which leadto LO/TO splitting were accounted for during the calcu-lations and the phonon acoustic sum rule was enforced.

Fourier interpolation was used to generate phononsalong high-symmetry directions for the production ofdispersion curves, as well as for 6000 wave vectors dis-tributed randomly throughout the Brillouin zone for theproduction of densities of states. As shown in Table II,Γ-point frequencies are within a few percent of values ob-tained through other calculations and experiments. Thefinal phonon dispersion curves shown in Figure 4 are alsoa close match for dispersion curves obtained previouslythrough DFT and inelastic neutron scattering.13,15

B. Gruneisen parameter calculations

The Gruneisen parameter relates the vibrational spec-trum of a material to its thermal expansion behaviour.At the macroscopic level, the volumetric coefficient ofthermal expansion of a material, αV, can be written as20

αV =γCV

BV(1)

where γ is the macroscopic Gruneisen parameter, CV isthe heat capacity at constant volume, B is the bulk mod-ulus and V is the system volume. Since CV , B and V areconstrained to have positive values, the sign of the ther-mal expansion coefficient αV is determined by the sign ofγ.

At the microscopic level, the mode Gruneisen param-

eter, γi,k, is given by20

γi,k = − V

ωi,k

∂ωi,k

∂V(2)

where V is the unit cell volume, ω the mode frequency,and the indices i and k refer to an individual phononby mode and wave vector respectively. γi,k is normallypositive for a given mode, since atomic bonds normallystiffen under compression, thereby increasing phonon fre-quency. However in some situations, such as for tension-effect modes as described in the Introduction and Figure2, mode frequency decreases on compression and thusγi,k is negative.γ is calculated by taking the sum of microscopic mode

Gruneisen parameter values, γi,k, weighted according tothe contribution of each mode to overall heat capacityCV .20 Therefore, phonons with positive γi,k contributeto PTE and phonons with negative γi,k contribute toNTE.

Values of γi,k were calculated for the a = 4.358 Aequilibrium structure by considering changes in phononfrequency and volume at a = 4.30 A. The result was thenconverted to a linear colour scale that ranged from red(γi,k ≤ −8) to white (γi,k = 0) through to blue (γi,k ≥+8).

Plotted dispersion curves were then shaded accordingto their corresponding value of γi,k on this scale, whilstbins that made up the plotted density of states wereshaded according to the average γi,k for each bin usingthe same colour scale. As can be seen in Figure 4, thisprocess allowed for easy identification and comparison ofdifferent modes’ contributions to positive and negativethermal expansion.

C. Generation of flexibility models

Simple models were devised to investigate flexibility inthe Cu2O framework. Our intention was not to reproduceany other properties of the system other than to abstractthe rigid and flexible parts of the framework down tovery stiff or complete flexibility. This approach allowedthe simulation of different types of tension effect in thismaterial.

Our flexibility models were created in the followingmanner. The equilibrium Cu2O cell was reconstructed

4

(a) (b) (c) (d)

FIG. 3. (a) The Cu2O structure in ball-and-stick representation: red atoms are O, blue atoms are Cu. (b) Representation ofCu2O Flexibility Model 1, consisting of corner-sharing rigid OCu4 tetrahedra. (c) Representation of Cu2O Flexibility Model2, consisting of rigid O–Cu–O rods: grey spheres represent flexible linkages. (d) Representation of Cu2O Flexibility Model 3,consisting of rigid Cu–O rods: grey spheres represent flexible linkages.

in the program GULP.42 A two-body harmonic potentialwas applied to the Cu–O bond and three-body harmonicpotentials were applied to Cu–O–Cu and O–Cu–O bonds.All atoms were assigned zero charge whilst equilibriumbond lengths and angles were chosen such that the cellwould remain unchanged upon a zero pressure geometryoptimisation.

We were able to control the rigid and flexible regions ofthe model by changing the force constants in this setup.For this study the force constant for the two-body po-tential was fixed at 100 eV A−2 to model a stiff Cu–Obond. The force constants for the three-body potentialswere set to either 100 eV rad−2 or zero, depending onwhether the flexibility model in question required thosebond angles to be fixed or to vary freely.

Flexibility models considered were:

1. Rigid OCu4 tetrahedra wherein Cu–O–Cu bond an-gles were fixed whilst O–Cu–O bond angles couldvary freely.

2. Rigid CuO2 rods wherein O–Cu–O bond angleswere fixed whilst Cu–O–Cu bond angles could varyfreely.

3. Rigid CuO rods wherein both O–Cu–O and Cu–O–Cu bond angles could both vary freely.

All of these models are illustrated in Figure 3. It can beseen in this diagram that Model 1 corresponds to rigidOCu4 tetrahedra; Model 2 to rigid O–Cu–O rods andModel 3 to rigid Cu–O rods. In addition, it should benoted that Model 1 and Model 2 are constrained versionsof Model 3. An even further constrained model, whereboth O–Cu–O and Cu–O–Cu bond angles were fixed, wasdiscounted as this would not have any free deformationsand thus could not correspond to any type of tensioneffect.

Phonons were calculated for all three flexibility modelsat the same wave vectors as in the ab-initio dispersioncurves and densities of states. Zero-frequency solutionscorresponded to modes involving free deformations of themodel structure, whilst solutions with non-zero frequency

corresponded to modes that violated the constraints ofthat model (e.g. Cu–O–Cu bond bending in Model 1).

D. Mapping of flexibility model phonons ontoab-initio phonons

For each ab-initio phonon mode at each wave vector a‘match’ value, mi,k, was defined as

mi,k =∑j

ei,k · ej,kΩ2 + ω2

j,k

(3)

where e is an eigenvector, ω is a phonon frequency, theindex i denotes a mode in the ab-initio calculation, theindex j denotes a mode in a given flexibility model andthe index k denotes a phonon wave vector. Ω is an arbi-trary scale factor that helps avoid divide-by-zero errors;we set it equal to 1 THz so that a ω2

j,k value of 0 gave a

1/(Ω2 + ω2j,k) value of 1 THz−2.

mi,k represents the degree to which the flexibilitymodel in question is able to reproduce the mode i atthe wave vector k. Scaling by Ω2 + ω2

j,k ensured thatonly flexibility model modes with zero or close-to-zerofrequency contributed to mi,k, as discussed in SectionII C above.

Due to the normalisation and orthogonality of theeigenvectors, each mi,k has a value between 0 and 1. Avalue of 0 indicates that the flexibility model could notreproduce the ab-initio mode eigenvector i at all, whilsta value of 1 indicates a perfect match.

A full set of mi,k were computed for the equilibriumCu2O structure. As with the γi,k plots, mi,k values werethen converted to a linear colour scale. This ranged fromwhite (mi,k = 0) through to black (mi,k = 1). Plottedab-initio dispersion curves were shaded according to theircorresponding value of mi,k and densities of states wereshaded according to the average mi,k for each bin.

As seen in Figure 4, this approach yields a convenientvisual representation of which real system phonons arereproduced by each flexibility model. This highlights

5

TABLE III. Cu2O equilibrium elastic and bulk moduli, as obtained through calculation and experiment. In order to account forscatter present in the elastic modulus data as calculated from acoustic modes close to the Γ point, values and errors presentedhere are interpolated from a straight line fit through data points obtained from each volume. Moduli are in units of GPa.Calculated values are close to values obtained via experiment. The symbol * indicates that, whilst no B0 value was quoted inthe original paper, it was inferred here using Equation 4. The symbol † indicates that the experiments were performed at roomtemperature. The symbol ‡ indicates that the experiments were performed at 4 K.

Source c11 c12 c44 B0

This work (PBE+pseudopotentials) 114.5± 0.9 87± 3 7.2± 0.7 96± 4

PBE+pseudopotentials32 116 100 8 106

Inelastic neutron scattering43† 126.1± 1.2 108.6± 0.4 13.6± 11.4 114.4± 1.3*

Ultrasonic interferometry44† 122.88± 0.38 106.50± 0.71 12.10± 0.30 111.96± 0.27

Pulse echo45‡ 121 105 10.9 105.7

the different types of tension effect-related deformationpresent in the full phonon spectrum.

E. Elastic modulus calculations

Acoustic phonon modes with negative γi,k, present inCu2O,13,15 are indicative of elastic softening. Changesin the elastic moduli could therefore be tracked by mon-itoring the softening of acoustic modes as a function ofpressure.20

A dynamical matrix solely for the acoustic modes wasconstructed for all six simulated cells. Monte Carlo min-imisation was then used to fit elastic modulus tensor val-ues to both the acoustic dynamical matrix and to theoriginal ab-initio acoustic phonon frequencies. The bulkmodulus for the equilibrium cell was also calculated usingthe cubic cell formula46

B =c11 + 2c12

3(4)

where B is the bulk modulus and cij are the i,jth com-ponents of the elastic modulus tensor.

Elastic moduli and bulk moduli for the equilibriumcell, compiled in Table III, are a good match for otherdata obtained through theory and experiment.

III. RESULTS

A. Thermal expansion in Cu2O

Equilibrium cell phonons, their associated modeGruneisen parameters and flexibility model match valuesare all shown in Figure 4. The results are a close matchto previous calculations and experimental data.13–15 Thelargest contribution to NTE in the 0–5 THz range comesfrom the transverse acoustic modes in the Γ–X–M re-gion of reciprocal space (γi,k ∼ −9 for Γ to X and theirsurroundings, slowly rising towards −3 approaching M).There is also some small contribution from several opticmodes spanning the Brillouin zone in the 2–5 THz range

(γi,k ∼ −1). The largest contribution to PTE in the 0–5 THz range comes from the three acoustic modes aroundR (γi,k ∼ +5) There is also some very small contributionaround Γ from the longitudinal acoustic mode and a pairof optic modes (γi,k ∼ +2).

Model 1, consisting of rigid OCu4 tetrahedra, is a rea-sonable match for some NTE and PTE modes but notfor others, suggesting there is limited correlation betweenthermal expansion and modes available to this model.The main good match is for a single, nearly dispersion-less, weak NTE optic mode at ∼ 2 THz that existsthroughout most of the Brillouin zone. This is a mod-erate match to a RUM consisting of OCu4 tetrahedralrotations and has a particularly strong match in the Γ–R direction. The same optic mode is also a very weakmatch for Model 2 (rigid CuO2 rods) but a strong matchfor Model 3 (which is a less constrained version of Model1, in that is allows Cu–O–Cu bond bending). We con-clude that, away from the Γ-R region, the mode is forcedto undergo some Cu–O–Cu bond bending as a result ofconstraints of the Cu2O structure and connectivity aswell as phonon wave vector—this point is discussed ingreater depth in Section IV A. There is therefore somesmall contribution to NTE from rotations of rigid or near-rigid OCu4 tetrahedra.

Model 2, consisting of rigid O–Cu–O rods, is a strongmatch to the principal NTE modes highlighted in Figure4. The NTE in Cu2O is therefore primarily driven by vi-brations of rigid O–Cu–O rods. Weak NTE optic modesthat do not match Model 1 are also moderate matches forModel 2. Again, as these modes are also a strong matchfor Model 3, it follows that the remaining NTE modesare almost a match for Model 2, but are forced to un-dergo some O–Cu–O bond bending for the same reasonsas stated above for Model 1. There is therefore also somesmall contribution to NTE from motion of near-rigid O–Cu–O rods.

Finally Model 3, consisting of rigid CuO rods, dom-inates the low energy region of the phonon spectrum,confirming that Cu–O bond stretching is a high energyprocess. Since the NTE-driving modes described by thismodel can also be described by Model 1 or Model 2, the

6

Wave vector

Freq

uenc

y (T

Hz)

K KX XM MR R

4

8

12

16

20

Wave vector

Freq

uenc

y (T

Hz)

K KX XM MR R

4

8

12

16

20

Wave vector

Freq

uenc

y (T

Hz)

K KX XM MR R

4

8

12

16

20

0

Wave vector

Freq

uenc

y (T

Hz)

K KX XM MR R

4

8

12

16

20

Wave vector

Freq

uenc

y (T

Hz)

K KX XM MR R

4

8

12

16

20

(a) (b)

(c) (d) (e)

8

0

0

1

-8

i,k mi,k

FIG. 4. (a) Phonon dispersion curves and full Brillouin zone density of states for the equilibrium Cu2O cell. We use thenotation of Bradley and Cracknell16 to label special points in reciprocal space: Γ = (0, 0, 0), X = (0.5, 0, 0), M = (0.5, 0.5, 0),and R = (0.5, 0.5, 0.5). Eigenvector matching was employed to differentiate crossings and anti-crossings in the dispersioncurves. (b) Shows the same data shaded according to the value of γi,k of each mode at each wave vector with respect tophonons calculated for a Cu2O cell of a = 4.30 A. The colour scale ranges from red (γi,k ≤ −8) to white (γi,k = 0) through toblue (γi,k ≥ +8). Bins that make up the density of states are shaded according to the average γi,k for each bin using the samecolour scale. (c), (d) and (e) Show the same data shaded according to the value of mi,k of each mode at each wave vector. Thecolour scale ranges from white (mi,k = 0) through to black (mi,k = 1). Bins that make up the densities of states are shadedaccording to the average value of mi,k for each bin using the same colour scale. (c) Shows mi,k values for Flexibility Model 1,(d) Shows mi,k values for Flexibility Model 2, (e) shows mi,k values for Flexibility Model 3.

extra flexibility afforded to Model 3 (compared the oth-ers) does not offer any additional tension effect that leadsto NTE in Cu2O.

Putting everything together, we can describe the ther-mal expansion of Cu2O in terms of its vibrational spec-trum as follows: the 0–1 THz range is dominated bystrong NTE modes in the Γ-X-M region of the Brillouinzone. These correspond to vibrations of rigid O–Cu–Orods. At ∼ 1.8 THz there is a strong PTE contributionfrom modes at and around the R point. This is expectedfor structures consisting of a pair of interpenetrating lat-tices: the R point corresponds to Γ for an individualsublattice, and acoustic modes here correspond to thesublattices moving in anti-phase.

At ∼ 2 THz there is a nearly dispersionless weak NTEmode that spans the entire Brillouin zone and thus formsthe large and sharp peak in the low energy phonon den-sity of states. This mode corresponds to rotations of rigidOCu4 tetrahedra where wave vector allows and to near-rigid OCu4 tetrahedra where constraints of frameworkconnectivity and wave vector mean rigid unit rotations

are not possible. Weak PTE modes around Γ also existin this frequency range; these correspond to rigid Cu–Orod motion and no particular tension effect.

At ∼ 3–4 THz there is a second band of weak NTEmodes that also span much of the Brillouin zone, butwith a greater dependence of frequency on wave vectorthan the ∼ 2 THz mode. These modes correspond tomotions of near-rigid O–Cu–O rods.

Higher frequency modes involve a non-trivial level ofCu–O bond stretching and thus contribute to standardPTE.

B. A predicted phase transition

Softening of the NTE phonon modes as a function ofpressure is illustrated in Figure 5. Zone-centre softeningof acoustic branches along Γ–X corresponds to instabili-ties in the elastic moduli c44 and c11 − c12 which would,in turn, drive a proper ferroelastic phase transition.20

Figure 6 shows elastic moduli for all six calculated vol-

7

0

Wave vector

Freq

uen

cy

(T

Hz)

K KX XM MR R

2

4

6

8

10

12

14

16

18

FIG. 5. Phonon dispersion curves and full Brillouin zone den-sity of states for Cu2O with unit cell parameter a = 4.25 Acorresponding to an effective pressure of 9.6 GPa. We usethe notation of Bradley and Cracknell16 to label specialpoints in reciprocal space: Γ = (0, 0, 0), X = (0.5, 0, 0),M = (0.5, 0.5, 0), and R = (0.5, 0.5, 0.5). Eigenvector match-ing was employed to differentiate crossings and anti-crossingsin the dispersion curves. The Γ–X acoustic NTE mode iscompletely soft at this pressure.

150

100

50

Ela

stic

Mo

du

lus

(GP

a)

8642-2-4

Pressure (GPa)

c11

c12

c44

(c11–c12) / 2

FIG. 6. The elastic moduli of Cu2O as a function of pressure.Straight line fits have been applied to the data for each modu-lus. c44 and c11−c12 both soften as a function of pressure. c44reaches zero before c11 − c12, indicating a ferroelastic phasetransition to an orthorhombic structure at 6.0± 0.8 GPa.

umes as a function of effective pressure. The plot showsfaster softening in c44, indicating a transition to an or-thorhombic phase20 at 6.0± 0.8 GPa.

In order to characterise the new phase, another geom-etry optimisation was carried out on the a = 4.25 A unitcell under an applied hydrostatic pressure of 8.5 GPa andusing the convergence criteria detailed earlier in SectionII A. On this occasion, however, only P1 symmetry wasapplied (though 90 cell angles remained fixed). In ad-dition, one cell parameter was changed to 4.26 A andthe fractional coordinate of O at (0, 0, 0) was changed

to (0.02, 0, 0). These minor adjustments ensured the ge-ometry optimiser did not become trapped at an energymaximum.

The relaxed cell was confirmed as orthorhombic withspace group Pnnn. Fractional coordinates of the cellcontents remained unchanged from their values in thecubic cell, however the cell parameters at 8.5 GPa becamea = 4.256 A, b = 4.261 A and c = 4.263 A.

IV. DISCUSSION

A. Framework flexibility and NTE in Cu2O

We now understand the driving force behind NTE inCu2O. It is dominated by low frequency rigid O–Cu–Orod motion, which corresponds to the translational acous-tic modes spanning the Γ–X–M region of reciprocal space.There is some small additional contribution at higherfrequencies from rigid and near-rigid OCu4 tetrahedralrotation as well as near-rigid O–Cu–O rod motion, allcorresponding to optic modes throughout the Brillouinzone.

The O–Cu–O rigid rod mechanism goes beyond thetraditional picture of a tension effect. Instead of trans-verse vibrations of individual bonds (as illustrated ear-lier in Figure 2) the mechanism consists of rotations ofrigid rods made up of multiple atoms. As these rods ro-tate they necessarily pull atomic planes together and thismotion drives macroscopic NTE.

The NTE in Cu2O can therefore be described as be-ing driven by RUMs, but with the rigid unit recast asan O–Cu–O rod. The additional NTE modes would thenbe O–Cu–O quasi-RUMs23 (QRUMs—modes close to be-ing RUMs but which involve some small deformation ofa rigid unit due to constraints of framework structure,connectivity and of wave vector), as well as both RUMsand QRUMs of an OCu4 tetrahedral unit.

That NTE in Cu2O is primarily driven by a modethat involves significant deformation of OCu4 tetrahedrahad previously been proposed.11,29,31 However, this studymarks the first time that any proposed mechanisms havebeen tested against eigenvectors of confirmed NTE modesthroughout the Brillouin zone. Furthermore, these pre-viously proposed mechanisms all assumed a Cu–O bondas the basic rigid unit; our work instead shows that anO–Cu–O rigid rod is in fact the fundamental rigid unitas far as NTE behaviour in this material is concerned.

Our analysis of the vibrational spectrum therefore alsoprovides insight into the intrinsic framework flexibility ofCu2O. As known previously,29 Cu–O bond stretching isa high energy process and this is also evident in our anal-ysis, as Cu–O bond stretching modes have been shownto exist only at frequencies greater than 4 THz.

The lowest energy deformations (0–1 THz) of thestructure are, however, RUMs involving rigid O–Cu–Orods. O–Cu–O bond angle bending does not occur un-til we reach optic modes at slightly higher frequencies

8

≥ 2 THz. There is therefore some small, but not trivial,cost to bending the O–Cu–O bond. This is, perhaps, notsurprising given the particularly high sensitivity to bondgeometry associated with bonds involving d orbitals—asis the case for the Cu centre of the O–Cu–O rod.

Meanwhile, one ∼ 2 THz optic mode consists of eitherOCu4 RUMs (along Γ–R) or OCu4 QRUMs (the restof the Brillouin zone), again as a result of constraintsof the framework structure and connectivity as well aswave vector. Cu–O–Cu bond bending has minimal effecton the mode frequency—hence the size of the peak inthe phonon density of states at this frequency. There istherefore minimal energy cost to Cu–O–Cu bond bend-ing.

B. Comparison with other NTE materials

The dominant NTE mode in Cu2O is reminiscent of themechanism driving weak low-temperature NTE in simpleframeworks with a diamond or zincblende lattice such asSi, Ga and CuCl.47 NTE in these materials is driven bytranslational acoustic vibrations exactly like those of therigid O–Cu–O rod model. This is not surprising giventhat Model 2 is effectively a pair of interpenetrating di-amondoid lattices. Unlike the aforementioned materialshowever, Cu2O has greater structural flexibility due tothe possibility of bending the O–Cu–O bond and thushas additional, albeit weak, NTE mechanisms availableto it.

Zn(CN)2, another material composed of two interpene-trating diamondoid lattices, might therefore be expectedto behave in a similar fashion to Cu2O. Like Cu2O,NTE in Zn(CN)2 is driven primarily by transverse acous-tic modes with some smaller contribution from opticmodes.19 However, whilst the acoustic modes in ques-tion are reminiscent of the O–Cu–O acoustic modes inCu2O (in that they minimise deformation of the Zn–N–C–Zn rod), in practice some minimal deformation of therod must take place as deformation of the Zn(C/N)4tetrahedral unit in Zn(CN)2 is a high frequency pro-cess. In addition, the dominant NTE-driving acous-tic mode in Zn(CN)2 spans the entire Brillouin zone,whilst the dominant acoustic NTE mode in Cu2O onlyspans the Γ–X–M region of reciprocal space. This, inturn, means that the NTE in Zn(CN)2 is much larger48

(αlinear = −16.9 MK−1), a difference that can be at-tributed to the CN bridges in Zn(CN)2 giving its frame-work more overall degrees of freedom than Cu2O.49

C. On the use of flexibility model mapping

The approach taken in this study, whereby phononeigenvectors generated from simple flexibility models aremapped onto full ab-initio calculated modes, has enabledidentification of modes in Cu2O that correspond to dif-ferent types of structural deformation. This, in turn, has

allowed for insightful evaluation of proposed NTE mech-anisms.

We have found this approach to be more efficientand less error-prone than relying on visual inspection ofmodes of interest. It has proven useful, in particular,when differentiating Cu2O modes that have very differ-ent features but exist at similar energies; such as in thecrowded 2–5 THz range of the phonon spectrum.

We anticipate that this same approach will provehighly useful in the analysis of framework flexibility andtension effect-driven NTE in more complex systems suchas metal-organic frameworks.50 These materials, a num-ber of which exhibit NTE,51,52 have unit cells typicallycomprising hundreds of atoms in addition to complex in-ternal bonding. As a result, a full analysis of their dis-persion curves using more conventional methods wouldbe a highly complex process.

D. High-pressure phase transitions

Given that negative Gruneisen parameter indicatesmode softening as a function of pressure,20 one could ar-gue that NTE itself is an outcome of the drive towards adisplacive phase transition at pressure. In this case, ourresults predict a transition to an orthorhombic phase at6.0± 0.8 GPa driven by the strongest NTE modes at Γ–X. However, in practice, there are numerous conflictingreports of high-pressure phase transitions of Cu2O.

At low temperatures Cu2O is in fact metastable above∼ 4 GPa,53,54 decomposing at equilibrium to form CuOand Cu. Some have predicted a displacive phase transi-tion on the basis of the existence of phonon soft modes55

as well as measured44 and calculated32 softening in c44and c11 − c12. Ab-initio calculations on strained Cu2Ocells32 are a close match to our results and show c44 soft-ening fastest, predicting a shear instability at ∼ 8 GPa.A transition was observed experimentally at 5 GPa55 us-ing X-ray powder diffraction and a diamond anvil cell.However it was not possible to conclusively identify thenew phase at that time.

On the other hand, further X-ray diffraction diamondanvil cell experiments have found no displacive phasetransition in the 0–10 GPa range. In one study56 thecubic structure was observed as remaining intact until aknown higher-pressure phase transition to a hexagonalstructure at 10 GPa. In another study53 the sample wasobserved to amorphise at pressure before transforming tothe same hexagonal phase at 11 GPa.

Another set of similar experiments did find a displacivetransition, but at 0.657 or 0.7–2.2 GPa,54 much smallerpressures than our prediction. This transition was ob-served to be proper ferroelastic in nature and the newphase was identified as tetragonal54 as opposed to ourprediction of an orthorhombic phase. Curiously, thestructure was also observed to transform into a pseu-docubic phase at 8.5 GPa.54

These inconsistencies can partly be attributed to the

9

fact that phase transitions observed experimentally inCu2O at high pressure are known to be highly sensitiveto the experimental environment, especially the hydro-staticity of the pressure medium.54,55 Furthermore signif-icant peak broadening under pressure obscures any peaksplitting,54 making pressure-induced displacive transi-tions in Cu2O difficult to observe experimentally.

Another factor which can complicate the identificationof the new phase is the fact that changes in the unitcell for displacive transitions are usually minute. In ourpredicted orthorhombic phase, the b and c parameterswere found to be especially close, within 0.05% of oneanother. Such a minute difference may explain why atetragonal phase is sometimes observed in experimentswhilst we predict an orthorhombic structure.

V. CONCLUSIONS

A full analysis of the NTE mechanism in Cu2O wasconducted. This was achieved by mapping differentstructural flexibility models onto results from ab-initiolattice dynamics calculations. The degree to which eachmodel could match the ab-initio phonons determined howwell it could describe specific modes in the real system.

It was found that NTE in Cu2O is dominated by lowfrequency vibrations of rigid O–Cu–O rods. There is alsosome smaller contribution at higher frequency from rigidand near-rigid OCu4 tetrahedral rotation and near-rigidO–Cu–O rod motion.

It was also found that the primary NTE mode drivesa proper ferroelastic phase transition at high pressure.Our simulations predict this to be to an orthorhombicstructure with space group Pnnn.

ACKNOWLEDGMENTS

LHNR is supported by NERC and CrystalMakerSoftware Ltd. ALG is supported by EPSRC(EP/G004528/2) and ERC (279705). Via our member-ship of the UK’s HPC Materials Chemistry Consortium,which is funded by EPSRC (EP/F067496), this workmade use of the facilities of HECToR, the UK’s nationalhigh-performance computing service, which is providedby UoE HPCx Ltd at the University of Edinburgh, CrayInc and NAG Ltd, and funded by the Office of Scienceand Technology through EPSRC’s High End Comput-ing Programme. Additional HECToR calculations wereperformed via membership of the UK Car-Parrinello con-sortium, which is funded by EPSRC (EP/F036884/1).

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